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A238130 Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.

%I A238130 #16 Jan 12 2015 04:12:38
%S A238130 1,1,0,2,0,0,2,2,0,0,3,4,1,0,0,2,10,4,0,0,0,4,12,14,2,0,0,0,2,22,29,
%T A238130 10,1,0,0,0,4,26,56,36,6,0,0,0,0,3,34,100,86,31,2,0,0,0,0,4,44,148,
%U A238130 200,99,16,1,0,0,0,0,2,54,230,374,278,78,8,0,0,0,0,0,6,58,322,680,654,274,52,2,0,0,0,0,0
%N A238130 Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.
%C A238130 First column (k=0) is A000005, second column (k=1) is 2*A002133.
%C A238130 Row sums are A011782.
%H A238130 Joerg Arndt and Alois P. Heinz, <a href="/A238130/b238130.txt">Rows n = 0..140, flattened</a>
%e A238130 Triangle starts:
%e A238130 00:  1,
%e A238130 01:  1, 0,
%e A238130 02:  2, 0, 0,
%e A238130 03:  2, 2, 0, 0,
%e A238130 04:  3, 4, 1, 0, 0,
%e A238130 05:  2, 10, 4, 0, 0, 0,
%e A238130 06:  4, 12, 14, 2, 0, 0, 0,
%e A238130 07:  2, 22, 29, 10, 1, 0, 0, 0,
%e A238130 08:  4, 26, 56, 36, 6, 0, 0, 0, 0,
%e A238130 09:  3, 34, 100, 86, 31, 2, 0, 0, 0, 0,
%e A238130 10:  4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0,
%e A238130 11:  2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0,
%e A238130 12:  6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0,
%e A238130 13:  2, 74, 446, 1122, 1390, 814, 225, 22, 1, 0, 0, 0, 0, 0,
%e A238130 ...
%e A238130 Row 5 is [2, 10, 4, 0, 0, 0] because in the 16 compositions of 5
%e A238130 ##:  [composition]  no. of changes
%e A238130 01:  [ 1 1 1 1 1 ]   0
%e A238130 02:  [ 1 1 1 2 ]   1
%e A238130 03:  [ 1 1 2 1 ]   2
%e A238130 04:  [ 1 1 3 ]   1
%e A238130 05:  [ 1 2 1 1 ]   2
%e A238130 06:  [ 1 2 2 ]   1
%e A238130 07:  [ 1 3 1 ]   2
%e A238130 08:  [ 1 4 ]   1
%e A238130 09:  [ 2 1 1 1 ]   1
%e A238130 10:  [ 2 1 2 ]   2
%e A238130 11:  [ 2 2 1 ]   1
%e A238130 12:  [ 2 3 ]   1
%e A238130 13:  [ 3 1 1 ]   1
%e A238130 14:  [ 3 2 ]   1
%e A238130 15:  [ 4 1 ]   1
%e A238130 16:  [ 5 ]   0
%e A238130 there are 2 with no changes, 10 with one change, and 4 with two changes.
%p A238130 b:= proc(n, v) option remember; `if`(n=0, 1, expand(
%p A238130       add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
%p A238130     end:
%p A238130 T:= n-> seq(coeff(b(n, 0), x, i), i=0..n):
%p A238130 seq(T(n), n=0..14);
%t A238130 b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]; T[n_] := Table[Coefficient[b[n, 0], x, i], {i, 0, n}]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 12 2015, translated from Maple *)
%Y A238130 Cf. A238279 (same sequence with zeros omitted).
%Y A238130 Cf. A106356 (compositions with k successive parts same).
%Y A238130 Cf. A225084 (compositions with maximal up-step k).
%K A238130 nonn,tabl
%O A238130 0,4
%A A238130 _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014